Mathematical Analysis of Reaction–Diffusion Equations Modeling the Michaelis–Menten Kinetics in a Micro-Disk Biosensor
In this study, we have investigated the mathematical model of an immobilized enzyme system that follows the Michaelis–Menten (MM) kinetics for a micro-disk biosensor. The film reaction model under steady state conditions is transformed into a couple differential equations which are based on dimensio...
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MDPI AG
2021-12-01
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| Series: | Molecules |
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| Online Access: | https://www.mdpi.com/1420-3049/26/23/7310 |
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| author | Naveed Ahmad Khan Fahad Sameer Alshammari Carlos Andrés Tavera Romero Muhammad Sulaiman Ghaylen Laouini |
| author_facet | Naveed Ahmad Khan Fahad Sameer Alshammari Carlos Andrés Tavera Romero Muhammad Sulaiman Ghaylen Laouini |
| author_sort | Naveed Ahmad Khan |
| collection | DOAJ |
| description | In this study, we have investigated the mathematical model of an immobilized enzyme system that follows the Michaelis–Menten (MM) kinetics for a micro-disk biosensor. The film reaction model under steady state conditions is transformed into a couple differential equations which are based on dimensionless concentration of hydrogen peroxide with enzyme reaction <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></semantics></math></inline-formula> and substrate <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow></semantics></math></inline-formula> within the biosensor. The model is based on a reaction–diffusion equation which contains highly non-linear terms related to MM kinetics of the enzymatic reaction. Further, to calculate the effect of variations in parameters on the dimensionless concentration of substrate and hydrogen peroxide, we have strengthened the computational ability of neural network (NN) architecture by using a backpropagated Levenberg–Marquardt training (LMT) algorithm. NNs–LMT algorithm is a supervised machine learning for which the initial data set is generated by using MATLAB built in function known as “pdex4”. Furthermore, the data set is validated by the processing of the NNs–LMT algorithm to find the approximate solutions for different scenarios and cases of mathematical model of micro-disk biosensors. Absolute errors, curve fitting, error histograms, regression and complexity analysis further validate the accuracy and robustness of the technique. |
| first_indexed | 2024-03-10T04:47:49Z |
| format | Article |
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| issn | 1420-3049 |
| language | English |
| last_indexed | 2024-03-10T04:47:49Z |
| publishDate | 2021-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Molecules |
| spelling | doaj.art-d86099ef733149bf8d2e72af156ec1a02023-11-23T02:50:34ZengMDPI AGMolecules1420-30492021-12-012623731010.3390/molecules26237310Mathematical Analysis of Reaction–Diffusion Equations Modeling the Michaelis–Menten Kinetics in a Micro-Disk BiosensorNaveed Ahmad Khan0Fahad Sameer Alshammari1Carlos Andrés Tavera Romero2Muhammad Sulaiman3Ghaylen Laouini4Department of Mathematics, Abdul Wali Khan University, Mardan 23200, PakistanDepartment of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi ArabiaCOMBA R&D Laboratory, Faculty of Engineering, Universidad Santiago de Cali, Cali 76001, ColombiaDepartment of Mathematics, Abdul Wali Khan University, Mardan 23200, PakistanCollege of Engineering and Technology, American University of the Middle East, Egaila 54200, KuwaitIn this study, we have investigated the mathematical model of an immobilized enzyme system that follows the Michaelis–Menten (MM) kinetics for a micro-disk biosensor. The film reaction model under steady state conditions is transformed into a couple differential equations which are based on dimensionless concentration of hydrogen peroxide with enzyme reaction <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></semantics></math></inline-formula> and substrate <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow></semantics></math></inline-formula> within the biosensor. The model is based on a reaction–diffusion equation which contains highly non-linear terms related to MM kinetics of the enzymatic reaction. Further, to calculate the effect of variations in parameters on the dimensionless concentration of substrate and hydrogen peroxide, we have strengthened the computational ability of neural network (NN) architecture by using a backpropagated Levenberg–Marquardt training (LMT) algorithm. NNs–LMT algorithm is a supervised machine learning for which the initial data set is generated by using MATLAB built in function known as “pdex4”. Furthermore, the data set is validated by the processing of the NNs–LMT algorithm to find the approximate solutions for different scenarios and cases of mathematical model of micro-disk biosensors. Absolute errors, curve fitting, error histograms, regression and complexity analysis further validate the accuracy and robustness of the technique.https://www.mdpi.com/1420-3049/26/23/7310micro-disk biosensormathematical modelingMichaelis–Menten kineticsenzymatic reactionartificial neural networkssoft computing |
| spellingShingle | Naveed Ahmad Khan Fahad Sameer Alshammari Carlos Andrés Tavera Romero Muhammad Sulaiman Ghaylen Laouini Mathematical Analysis of Reaction–Diffusion Equations Modeling the Michaelis–Menten Kinetics in a Micro-Disk Biosensor Molecules micro-disk biosensor mathematical modeling Michaelis–Menten kinetics enzymatic reaction artificial neural networks soft computing |
| title | Mathematical Analysis of Reaction–Diffusion Equations Modeling the Michaelis–Menten Kinetics in a Micro-Disk Biosensor |
| title_full | Mathematical Analysis of Reaction–Diffusion Equations Modeling the Michaelis–Menten Kinetics in a Micro-Disk Biosensor |
| title_fullStr | Mathematical Analysis of Reaction–Diffusion Equations Modeling the Michaelis–Menten Kinetics in a Micro-Disk Biosensor |
| title_full_unstemmed | Mathematical Analysis of Reaction–Diffusion Equations Modeling the Michaelis–Menten Kinetics in a Micro-Disk Biosensor |
| title_short | Mathematical Analysis of Reaction–Diffusion Equations Modeling the Michaelis–Menten Kinetics in a Micro-Disk Biosensor |
| title_sort | mathematical analysis of reaction diffusion equations modeling the michaelis menten kinetics in a micro disk biosensor |
| topic | micro-disk biosensor mathematical modeling Michaelis–Menten kinetics enzymatic reaction artificial neural networks soft computing |
| url | https://www.mdpi.com/1420-3049/26/23/7310 |
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